Quasi-static crack growth in materials displaying the Bauschinger effect—I. Steady-state analysis

Asymptotic solution for mode III crack growth in J 2-elastoplasticity with mixed isotropic-kinematic strain hardening

“…It can be observed that the material particle is in a stress-free state when it lies at an infinite distance ahead of the crack tip (0 = 0), it experiences almost proportional loading before = 7r/4, then it suffers a strongly non proportional stress history and, finally, it approaches a finite value of stress at infinite distance behind the crack tip, for 0 = 7r (this tendency to a finite value of stress on the crack flanks is clearly elucidated in [19], and the same arguments apply in the present context). If we consider that our asymptotic solution is known except for an amplitude factor, Figure 9 shows impressive agreement with the numerical finite element results of [15] (their Figure 4).…”

“…The results obtained are consistent with the finite element simulation of mode III crack growth of Narasimhan et al [ 15] and show that the kinematical component of hardening plays an important role on the crack-tip field. Among other effects of the hardening anisotropy, we note that the solution tends to a solution corresponding to a crack steadily propagating in an elastic medium in the limiting case of pure kinematic hardening.…”

“…The main differences in the stress distribution occur ahead of the crack tip, in the sector between 0 ° and 90 °, where, for kinematic hardening, a decrease of T~ with 0, corresponding to an increase of Tr, is observed. Moreover, when 0 tends to z-, T~ approaches a negative value, as noted in [8] for isotropic hardening and in [15] for kinematic hardening. There are however other effects related to an increase in the kinematic component of hardening.…”

“…In contrast to the isotropic hardening case, when b is small enough continued yielding occurs when the crack tip leaves the material point behind it. This effect was noted [15], using a finite element simulation of crack growth. Moreover, Lam and McMeeking [22] and Narasimhan et al [15] found a radial variation of the near crack-tip fields in the form of r s, which validates the assumption (15).…”

“…If we differentiate with respect to 0 both members of (16.2), (16.4), (16.5), and of (1), transformed with (15), and evaluate the resulting functions at 0 = 0, we obtain (26) and (27). It may be important to remark that (26-28) have been obtained under the hypothesis of non-singular behavior of angular functions at # = 0.…”